/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 100 Let \(Z\) be a standard normal r... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 100

Let \(Z\) be a standard normal random variable and let \(Y_{1}=Z\) and \(Y_{2}=Z^{2}\). a. What are \(E\left(Y_{1}\right)\) and \(E\left(Y_{2}\right) ?\) b. What is \(E\left(Y_{1} Y_{2}\right) ?\left[\text { Hint: } E\left(Y_{1} Y_{2}\right)=E\left(Z^{3}\right), \text { recall Exercise 4.199. }\right]\) c. What is \(\operatorname{Cov}\left(Y_{1}, Y_{2}\right) ?\) d. Notice that \(P\left(Y_{2}>1 | Y_{1}>1\right)=1 .\) Are \(Y_{1}\) and \(Y_{2}\) independent?

Short Answer

Expert verified
a. \(E(Y_1) = 0\), \(E(Y_2) = 1\); b. \(E(Y_1 Y_2) = 0\); c. \(\operatorname{Cov}(Y_1, Y_2) = 0\); d. No, they are not independent.

Step by step solution

01

Calculate E(Y_1)

The expectation of a standard normal random variable \(Z\) is zero. Thus, \(E(Y_1) = E(Z) = 0\).
02

Calculate E(Y_2)

Since \(Y_2 = Z^2\) and \(Z\) is a standard normal random variable, the expectation of \(Z^2\) is 1, as the variance of a standard normal distribution is 1. Therefore, \(E(Y_2) = E(Z^2) = 1\).
03

Calculate E(Y_1 Y_2)

We are given that \(E(Y_1 Y_2) = E(Z^3)\). Since \(Z\) is symmetric about zero and \(Z^3\) is an odd function, \(E(Z^3) = 0\). Hence, \(E(Y_1 Y_2) = 0\).
04

Calculate Cov(Y_1, Y_2)

The covariance is given by \(\operatorname{Cov}(Y_1, Y_2) = E(Y_1 Y_2) - E(Y_1)E(Y_2)\). Substituting the values from the previous steps gives \(\operatorname{Cov}(Y_1, Y_2) = 0 - (0)(1) = 0\).
05

Assess Independence of Y_1 and Y_2

For two variables to be independent, \(P(Y_2 > 1 | Y_1 > 1)\) must not always equal 1 simply because their relationship depends on the condition. Here, \(P(Y_2 > 1 | Y_1 > 1) = 1\) suggests that \(Y_1\) and \(Y_2\) are not independent, even though their covariance is 0, indicating a non-linear dependence.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Normal Distribution
The standard normal distribution is one of the most fundamental concepts in statistics. It is a normal distribution with a mean of zero and a variance of one. This means the bell-shaped curve is centered at zero and is spread out in such a way that the standard deviation is also one. The variable that follows this distribution is typically denoted as \(Z\).

A special property of a standard normal variable is its symmetry around the mean. This symmetry leads to the fact that any odd function of \(Z\) such as \(Z^3\) has an expectation of zero (as the positive and negative parts cancel each other out over the distribution). This property is crucial when calculating expectations of functions of \(Z\).
  • Mean: \(0\).
  • Variance: \(1\).
  • Symmetric about zero.
Expectation
Expectation, in probability, refers to the expected value or the mean of a random variable. For a random variable like \(Z\) which follows a standard normal distribution, the expectation \(E(Z)\) is zero.

When dealing with functions of a random variable, we use properties of expectations to compute results. For instance, if you have \(Y_2 = Z^2\), the expectation becomes the sum of all probabilities multiplied by their respective outcomes. This is why \(E(Y_2)=1\) as it's the variance for a standard normal distribution (the squared outcomes balance each other to yield a mean that equals the variance).
  • Symbol: \(E(X)\).
  • \(E(Z) = 0\).
  • \(E(Z^2) = 1\).
Independence and Dependence in Probability
In probability theory, independence between two variables means that the occurrence of one doesn't influence the probability of the other. It's a critical concept when assessing the relationship between \(Y_1\) and \(Y_2\). Even if the covariance between two random variables is zero, they might not be independent. Zero covariance only implies no linear relationship.

In the exercise, observing that \(P(Y_2 > 1 | Y_1 > 1) = 1\) suggests \(Y_1 = Z\) and \(Y_2 = Z^2\) are not independent. This denotes a deterministic linkage where all occurrences of \(Y_1\) imply a certain outcome in \(Y_2\).
  • Covariance Zero 鈮� Independence.
  • Conditional probability can indicate dependence.
  • Non-linear relationships might not affect covariance.
Variance
Variance measures how much values of a random variable, like \(Z\), deviate from the mean. For \(Z\) in a standard normal distribution, the variance is defined as one. This single number shows the spread within the values and is calculated as \(E[(Z - E(Z))^2]\). Since \(E(Z) = 0\), the calculation simplifies to \(E(Z^2)\), which is also 1.

Variance helps in understanding the size of fluctuations around the mean, and for our variables \(Y_1 = Z\) and \(Y_2 = Z^2\), its role in analysis is telling. For example, the variance determines the expectation of \(Y_2\) since \(E(Y_2) = Variance(Z) = 1\), explaining the consistency and spread of \(Y_1\).
  • Symbol: \(Var(X)\).
  • Formula: \(E[(X - E(X))^2]\).
  • For \(Z\), \(Var(Z) = 1\).

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(Y_{1}\) and \(Y_{2}\) be independent random variables that are both uniformly distributed on the interval \((0,1) .\) Find \(P\left(Y_{1}<2 Y_{2} | Y_{1}<3 Y_{2}\right)\).

The weights of a population of mice fed on a certain diet since birth are assumed to be normally distributed with \(\mu=100\) and \(\sigma=20\) (measurement in grams). Suppose that a random sample of \(n=4\) mice is taken from this population. Find the probability that a. exactly two weigh between 80 and 100 grams and exactly one weighs more than 100 grams. b. all four mice weigh more than 100 grams.

The random variables \(Y_{1}\) and \(Y_{2}\) are such that \(E\left(Y_{1}\right)=4, E\left(Y_{2}\right)=-1, V\left(Y_{1}\right)=2\) and \(V\left(Y_{2}\right)=8\). a. What is \(\operatorname{Cov}\left(Y_{1}, Y_{2}\right) ?\) b. Assuming that the means and variances are correct, as given, is it possible that \(\operatorname{Cov}\left(Y_{1}, Y_{2}\right)=7 ?\left[\text { Hint: } \text { If } \operatorname{Cov}\left(Y_{1}, Y_{2}\right)=7, \text { what is the value of } \rho, \text { the coefficient of correlation? }\right]\) c. Assuming that the means and variances are correct, what is the largest possible value for \(\operatorname{Cov}\left(Y_{1}, Y_{2}\right) ?\) If \(\operatorname{Cov}\left(Y_{1}, Y_{2}\right)\) achieves this largest value, what does that imply about the relationship between \(Y_{1}\) and \(Y_{2} ?\) d. Assuming that the means and variances are correct, what is the smallest possible value for \(\operatorname{Cov}\left(Y_{1}, Y_{2}\right) ?\) If \(\operatorname{Cov}\left(Y_{1}, Y_{2}\right)\) achieves this smallest value, what does that imply about the relationship between \(Y_{1}\) and \(Y_{2} ?\)

In Exercise 5.9 , we determined that $$f\left(y_{1}, y_{2}\right)=\left\\{\begin{array}{ll} 6\left(1-y_{2}\right), & 0 \leq y_{1} \leq y_{2} \leq 1 \\ 0, & \text { elsewhere } \end{array}\right.$$ is a valid joint probability density function. Find a. the marginal density functions for \(Y_{1}\) and \(Y_{2}\) b. \(P\left(Y_{2} \leq 1 / 2 | Y_{1} \leq 3 / 4\right)\) c. the conditional density function of \(Y_{1}\) given \(Y_{2}=y_{2}\) d. the conditional density function of \(Y_{2}\) given \(Y_{1}=y_{1}\) e. \(P\left(Y_{2} \geq 3 / 4 | Y_{1}=1 / 2\right)\)

Suppose that the random variables \(Y_{1}\) and \(Y_{2}\) have joint probability density function \(f\left(y_{1}, y_{2}\right)\) given by $$f\left(y_{1}, y_{2}\right)=\left\\{\begin{array}{ll} 6 y_{1}^{2} y_{2}, & 0 \leq y_{1} \leq y_{2}, y_{1}+y_{2} \leq 2 \\ 0, & \text { elsewhere } \end{array}\right.$$ a. Verify that this is a valid joint density function. b. What is the probability that \(Y_{1}+Y_{2}\) is less than \(1 ?\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.